\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 179, pp. 1--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/179\hfil 3D regularized MHD equations]
{Analyticity of the global attractor for the 3D regularized MHD equations}

\author[C. Zhao, B. Li \hfil EJDE-2016/179\hfilneg]
{Caidi Zhao, Bei Li}

\address{Caidi Zhao \newline
Department of Mathematics and Information Science,
Wenzhou University, Wenzhou, Zhejiang 325035, China}
\email{zhaocaidi2013@163.com}

\address{Bei Li \newline
Department of Mathematics and Information Science,
Wenzhou University, Wenzhou, Zhejiang 325035, China}
\email{Lbbeili2015@163.com}

\thanks{Submitted November 10, 2015. Published July 7, 2016.}
\subjclass[2010]{76A10, 76D03, 35Q35}
\keywords{Regularized MHD equations; global attractor; Gevrey regularity;
\hfill\break\indent analyticity}

\begin{abstract}
 We study the three-dimensional (3D) regularized
 magnetohydrodynamics (MHD) equations. Using the method of splitting
 of the asymptotic approximate solutions into higher and lower
 Fourier components, we prove that the global attractor of the 3D
 regularized MHD equations consists of real analytic functions,
 whenever the forcing terms are analytic.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

In this article, we investigate the regularized
magnetohydrodynamics (MHD) equations
\begin{gather}
 \partial_t(u-\alpha^2\Delta u)
 -\nu \Delta u +(u\cdot \nabla)u
 -(b\cdot \nabla)b+\nabla p
 =f, \quad (x,t)\in \Omega\times \mathbb{R}_+, \label{1.1}\\
 \partial_t(b-\beta^2\Delta b)
 -\mu \Delta b +(u\cdot \nabla)b
 -(b\cdot \nabla)u
 =g, \quad (x,t)\in \Omega\times \mathbb{R}_+, \label{1.2}\\
 \nabla\cdot u
 =\nabla\cdot b =0, \quad (x, t)\in \Omega\times \mathbb{R}_+,\label{1.3}\\
 u(x,0) =u^{\rm in},\quad
 b(x,0) =b^{\rm in}, \quad x\in \Omega,\label{1.4}
\end{gather}
which are regularizations in both the velocity and the magnetic
field of the following large eddy simulation model for the turbulent
flow of a magnetofluid (see \cite{C11}):
\begin{gather}
 \partial_t u
 -\nu \Delta u +(u\cdot \nabla)u
 -(b\cdot \nabla)b+\nabla p
 =f, \quad (x,t)\in \Omega\times \mathbb{R}_+, \label{1.5}\\
 \partial_t b
 -\mu \Delta b +(u\cdot \nabla)b
 -(b\cdot \nabla)u
 =g, \quad (x,t)\in \Omega\times \mathbb{R}_+, \label{1.6}\\
 \nabla\cdot u =\nabla\cdot b
 =0, \quad (x, t)\in \Omega\times \mathbb{R}_+,\label{1.7}\\
 u(x,0)=u^{\rm in},\quad
 b(x,0)=b^{\rm in}, \quad x\in \Omega,\label{1.8}
\end{gather}
where the velocity field $u=(u_1, u_2, u_3)$, the magnetic field
$b=(b_1, b_2, b_3)$ and the total pressure $p(x,t)$ are the unknown
terms, $\nu$ is the kinematic viscosity and $\mu$ is the constant
magnetic resistivity, $f$ represents volume force applied to the
fluid, $g$ is usually zero when Maxwell's displacement currents are
ignored. We will assume the constants $\nu,\mu,\alpha$ and $\beta$
are all positive. We consider the case $\Omega=[0, L]^3\subset
\mathbb{R}^3$ ($L>0$) and assume space-periodic conditions on the
initial data so that the corresponding solutions are space-periodic.


Compared to equations \eqref{1.5} and \eqref{1.6}, equations
\eqref{1.1} and \eqref{1.2} contain the extra regularizing terms
$(-\alpha^2
\partial_t\Delta u)$ and $(-\beta^2
\partial_t\Delta b)$, respectively. These two terms have two main
effects. On the one hand, they regularize the equations in a way
that the 3D equations \eqref{1.1}-\eqref{1.4} become now globally
wellposed (see Lemma \ref{L2.2} in Section 2 or \cite{C11}). On the
other hand, they change the parabolic character of the MHD equations
to the regularized one. For this reason, we call equations
\eqref{1.1}-\eqref{1.2} as regularized MHD equations.

We can view equations \eqref{1.1}-\eqref{1.4}, also called
MHD-Voight equations, as the approximate equations for the MHD
equations \eqref{1.5}-\eqref{1.8} as $\alpha, \beta\to 0$.
Equations \eqref{1.1}-\eqref{1.4} are the exact equations for a
class of visco-elastic fluid known as Kelvin-Voight fluids (the
corresponding equations for nonmagnetic fluids were first introduced
by Oskolkov in \cite{O73}). Equations \eqref{1.5}-\eqref{1.8} (or
the relevant equations) have been widely studied, see e.g.
\cite{CS11, ST83} for the existence and uniqueness of the solutions
and \cite{CW10, W04, Zh07, ZG10} for the regularity criteria. Also,
equations \eqref{1.1}-\eqref{1.4} have been deeply studied, see
e.g., Catania and Secchi \cite{CS10, CS11}, Catania \cite{C11},
Larios and Titi \cite{LT10, LT14}, Levant, Ramos and Titi
\cite{LRT10}. Particularly, Catania \cite{C11} studied the global
attractor and determining modes for the solutions of equations
\eqref{1.1}-\eqref{1.4} in $[0, 2\pi L]^3$ ($L>0$) with
space-periodic conditions.


Attractor is an important concept in the study of
infinite-dimensional dynamical systems. There are some monographs
concerning this subject, see, e.g., \cite{CV02,R01,SY02,T97}. At the
same time, the theory and method within these monographs have been
extensively applied to many concrete partial differential equations
arising in mathematical physics. For instance, we can refer to
\cite{ZZ07,ZLZ09,ZLW14}.
We note that Foias, Manley, Rosa and Temam
 in the monograph \cite{FMRT04} researched the determining modes and
nodes, as well as the Gevrey regularity of the global attractor for
the Navier-Stokes equations. The Gevrey class regularity of the
attractor reveals that the solutions lying in the attractor are
analytic with values in a Gevrey class of analytic functions in
space.

Nowadays, there are some papers investigating the Gevrey class
regularity of the solutions. For example, we can refer to
\cite{B05,BS07,CRT99, FT89,L92} for the Navier-Stokes equations; to
\cite{YL05,YLH07} for the Navier-Stokes-$\alpha$ equations; to
\cite{PV11, PRP12} for the second-grade fluid equations; to
\cite{BS12} for a class of dissipative equations; to \cite{BT96} for
a class of hypoelliptic equations; to \cite{CH99} for the
time-dependent Ginzburg-Landau equations; to \cite{FT98} for
nonlinear analytic parabolic equations; to \cite{GK09} for a class
of water-wave models; to \cite{KV11,LO97} for the Euler equations;
to \cite{LT99,OT00} for the B\'enard equation; to \cite{M99} for the
laser equations; to \cite{OT98} for the weakly damped driven
nonlinear Schr\"odinger equation; to \cite{P01} for the
Kuramoto-Sivashinsky equation; to \cite{S09} for the micropolar
fluid equations, so on and so forth.


Recently, Levant and Titi proved in \cite{KLT09} the Gevrey
regularity for the attractor of the 3D Navier-Stokes-Voight
equations. The method of the proof in \cite{KLT09} is the splitting
of the velocity into higher and lower Fourier components. The method
of splitting of the unknown functions into higher and lower Fourier
components has been used before in the context of the weakly damped
driven nonlinear Schr\"odinger equation in Oliver and Titi
\cite{OT98} and a model of B\'eard convection in a porous medium in
Oliver and Titi \cite{OT00} (see also Goubet \cite{G96}). Recently,
the authors of Chueshov \emph{et al}. \cite{CPS04} followed the same
methods to prove the Gevrey regularity of the global attractor of
the generalized Benjamin-Bona-Mahony equation.

The purpose of this paper is to establish the Gevrey regularity of
the global attractor for the 3D MHD equations
\eqref{1.1}-\eqref{1.4}. Our result reveals that all solutions
within the global attractor are real analytic functions, whenever
the forcing terms are analytic.

We want to point that the idea of this paper originates from article
\cite{KLT09}. Different to the Navier-Stokes-Voight equations
studied in \cite{KLT09}, equations \eqref{1.1}-\eqref{1.4} contain
the coupled Maxwell's equations which rule the magnetic field.
Observing the coupled structure of the addressed equations, we
expect to extend the method of the proof in \cite{KLT09}--splitting
both the velocity field $u(x,t)$ and the magnetic field $b(x,t)$
into higher and lower Fourier components. In terms of these
splitting, we then construct the asymptotic approximations of the
solutions in the Sobolev space $V_m$ and the Gevrey space $G_\tau^1$
(see Section 2 for notations). When doing so, however, we need give
proper decomposition of the equations. At the same time, we find
that the coupled structure of the regularized MHD equations plays an
important role when we estimate the relevant nonlinear terms in the
asymptotic approximate solutions.

This article is organized as follows. In the next section, we introduce
some notations and operators, as well as some lemmas that will be
used frequently in our proof. In Section 3, we construct the
asymptotic approximation of the solution of the regularized MHD
equations in the Sobolev space $V_m$ for $m\geqslant 2$. In Section
4, we construct the asymptotic approximation of the solution in the
Gevrey space $G^2_\tau$ for some $\tau>0$.


\section{Preliminaries}

 Throughout this article, we denoted by $\mathbb{L}^p(\Omega)=L^p(\Omega)\times L^p(\Omega)\times L^p(\Omega)$, for
$1\leqslant p\leqslant \infty$, and
$\mathbb{H}^m(\Omega)=H^m(\Omega)\times H^m(\Omega)\times H^m(\Omega)$ the 3D
vector Lebesgue and Sobolev spaces (see \cite{A75}) of the periodic
functions on $\Omega$, respectively. Let $\digamma$ be the set of
all 3D vector trigonometric polynomials on the periodic domain
$\Omega$, and denote
\begin{gather*}
 \mathcal{V}= \big\{\phi\in \digamma
 : \nabla\cdot\phi=0 \,\,{\rm and} \int_\Omega\phi(x){\mathrm d}x=0
 \big\},\\
 H=\text{closure of $\mathcal{V}$ in the $\mathbb{L}^2(\Omega)$ topology},\\
 V=\text{closure of $\mathcal{V}$ in the $\mathbb{H}^1(\Omega)$ topology}.
\end{gather*}
Also, we let $P_\sigma : \mathbb{L}^2(\Omega)\to H$ be the Helmholtz-Leray projection operator and
 $A= -P_\sigma\Delta$ be the stokes operator subject
 to the periodic boundary conditions with
 domain $D(A)=\mathbb{H}^2(\Omega)\cap V$.
Notice that in the space-periodic case
\[
 Au=-P_\sigma\Delta u=-\Delta u, \quad \forall u\in D(A).
\]
We know that the operator $A^{-1}$ is a positive definite,
self-adjoint, compact operator from $H$ into $H$. Thus, for any
$s\in {\mathbb R}$, we can define the Hilbert space $V_s=D(A^{s/2})$
endowed with the inner product and norm as
\[
 (u, v)_s=\sum_{j\in {\mathbb Z}^3}(u_j\cdot v_j|j|^{2s}), \quad
 |u|^2_s=(u,u)_s,
\]
for any $u, v\in V_s$, where $u_j, v_j$ are the corresponding
Fourier coefficients of $u$ and $v$, respectively. Obviously, we
have $V=V_1$ and $V_0=H$. We will denote the inner product and norm
in $V_0=H$ as
\[
 (u, v)=\sum_{j\in {\mathbb Z}^3}(u_j\cdot v_j), \quad
 \|u\|^2=(u,u), \quad \forall u, v\in V_0.
\]

To deal with the convective terms in equations \eqref{1.1} and
\eqref{1.2}, we introduce the bilinear form
\[
 B(u, v)=P_\sigma((u\cdot\nabla)v), \quad \forall u, v\in \mathcal{V}.
\]
We see (e.g., \cite{KLT09, T01}) that $B$ can be extended to a
continuous map $B: V\times V\longmapsto V'$, where $V'=V_{-1}$ is
the dual space of $V$, moreover,
\begin{equation} \label{2.1}
 |\langle B(u, v), w\rangle_{V',V}|
\leqslant
 c(\Omega)\lambda_1^{-3/4}\|u\|^{1/2}|u|^{1/2}_1|v|_1|w|_1, \quad
 \forall u,v,w\in V,
\end{equation}
where $\lambda_1$ is the first eigenvalue of $A$, $\langle \cdot,
\cdot\rangle_{V',V}$ is the dual product between $V'$ and $V$, and
hereafter $c(\cdot)$ or $c(\cdot, \cdot, \cdots,\cdot)>0$ denotes
the generic constant depending only the quantities appearing in the
bracket, which may take different values from one line to the next.
Furthermore, we have

\begin{lemma}\label{L2.1}
The bilinear form $B(\cdot, \cdot)$ satisfies
\begin{itemize}

\item[(I)] If $u, b\in V$, then all $B(u,u)$, $B(b,b)$,
$B(u,b)$, $B(b,u)$ belong to $V_{-1/2}$, and
\begin{gather*}
 |B(u,u)|_{-1/2}
\leqslant c(\Omega)\lambda_1^{-3/4}|u|^2_1, \\
 |B(b,b)|_{-1/2}
\leqslant c(\Omega)\lambda_1^{-3/4}|b|^2_1, \\
 |B(u,b)|_{-1/2}
\leqslant c(\Omega)\lambda_1^{-3/4}|u|_1|b|_1, \\
 |B(b,u)|_{-1/2}
\leqslant c(\Omega)\lambda_1^{-3/4}|u|_1|b|_1.
 \end{gather*}

\item[(II)] If $u, b\in V_{3/2}$, then all $B(u,u)$, $B(b,b)$,
$B(u,b)$, $B(b,u)$ belong to $H$, and
\begin{gather*}
 \|B(u,u)\|
\leqslant c(\Omega)\lambda_1^{-3/4}|u|_1|u|_{3/2}, \\
 \|B(b,b)\|
\leqslant c(\Omega)\lambda_1^{-3/4} |b|_1|b|_{3/2}, \\
 \|B(u,b)\|
\leqslant c(\Omega)\lambda_1^{-3/4}|u|_1|b|_{3/2}, \\
 \|B(b,u)\|
\leqslant c(\Omega)\lambda_1^{-3/4}|u|_{3/2}|b|_1.
 \end{gather*}

\item[(III)] For any $m\geqslant 1$,
if $u, b\in V_{m+1}$, then all $B(u,u)$, $B(b,b)$, $B(u,b)$,
$B(b,u)$ belong to $V_m$, and
\begin{gather*}
 |B(u,u)|_m
\leqslant
 c(m,\Omega)\lambda_1^{-7/8}|u|^{1/4}_1|u|_{2}^{3/4}|u|_{m+1}, \\
 |B(b,b)|_m
\leqslant c(m,\Omega) \lambda_1^{-7/8}|b|^{1/4}_1|b|_{2}^{3/4}|b|_{m+1}, \\
 |B(u,b)|_m
\leqslant c(m,\Omega)\lambda_1^{-7/8}|u|^{1/4}_1|u|^{3/4}_{2}|b|_{m+1}, \\
 |B(b,u)|_m
\leqslant c(m,\Omega)\lambda_1^{-7/8}|b|^{1/4}_1|b|^{3/4}_{2}|u|_{m+1}.
 \end{gather*}
\end{itemize}
\end{lemma}

\begin{proof}
 The conclusions of this lemma can be obtained by the standard interpolation
 estimates and the Gagliardo-Nirenberg inequity (see e.g.,
 \cite{T01,T95}). Here we omit the details.
\end{proof}

With the above notation and definitions, we can write equations
\eqref{1.1}-\eqref{1.4} in the equivalent functional form
\begin{gather}
 u_t+\nu Au+\alpha^2 Au_t+B(u, u)-B(b,b)=f, \label{2.14}\\
 b_t+\mu Ab+\beta^2 Ab_t+B(u, b)-B(b,u)=g, \label{2.15}\\
 u(x,0)=u^{\rm in},\quad
 b(x,0)=b^{\rm in}. \label{2.16}
\end{gather}
For the global well-posedness of the solutions and existence of the
global attractor for equations \eqref{2.14}-\eqref{2.16}, we have

\begin{lemma}[\cite{C11}] \label{L2.2}
 Assume that the initial data and the forcing terms
$f$ and $g$ are space-periodic with zero-spatial-mean vector fields
that satisfy $f, g\in L^\infty({\mathbb R}_+; H)$, $(u^{\rm in},
b^{\rm in})\in V\times V$, and $\nabla\cdot
 u^{\rm in}=\nabla\cdot b^{\rm
in}=0$. Then equations \eqref{2.14}-\eqref{2.16} have a unique
global solution $(u, b)$ such that, for each $T>0$, one has
\[
 u, b\in L^\infty([0, T]; V)\cap L^2([0,T]; \mathbb{H}^2(\Omega)).
\]
If furthermore $f$ and $g$ are independent of time $t$, then the
semigroup $\{S(t)\}_{t\geqslant 0}$ generated by the solution
operators possesses a unique global attractor ${\mathcal A}\subset
V\times V$.
\end{lemma}

The motivation of this paper is to investigate the analyticity of
the solutions within the global attractor ${\mathcal A}$. We will
employ the concept of the Gevrey class regularity. For some given
$\tau>0$ and $r\geqslant 0$, we define the Gevrey space as
\[
 G_\tau^r
= D(A^{r/2}e^{\tau A^{1/2}})
= \big\{
 u\in H \big |\,
 \|A^{r/2}e^{\tau A^{1/2}} u\|^2
 =\sum_{j\in {\mathbb Z}^3}|u_j|^2|j|^{2r}e^{2\tau |j|}<\infty
 \big\},
\]
which is endowed with the inner product and norm as follows
\begin{gather*}
 (u, v)_{r, \tau}
= \Big(A^{r/2}e^{\tau A^{1/2}} u, A^{r/2}e^{\tau A^{1/2}} v \Big)
= \sum_{j\in {\mathbb Z}^3}u_j\cdot v_j|j|^{2r}e^{2\tau |j|}, \\
 |u|_{r, \tau}
=\|A^{r/2}e^{\tau A^{1/2}} u\|, \quad u, v\in G_\tau^r,
\end{gather*}
where $u_j$ and $v_j$ are the corresponding Fourier coefficients of
$u$ and $v$, respectively. Note that Levermore and Oliver
\cite{LO97} proved that the space of real analyticity functions
${\mathcal C}^{\omega}(\Omega)$ has the following characterization
\[
 {\mathcal C}^{\omega}(\Omega)=\cup_{\tau >0}G_{\tau}^r.
\]


We end this section with some technique lemmas that will be used in
the proof of our main results. First, let $\lambda>0$ and denote by
$P_\lambda$ the $H$-orthogonal projection onto the span of
eigenfunctions of $A$ corresponding to eigenvalues of the magnitude
less than or equal to $\lambda$. Set $Q_\lambda=I-P_\lambda$. The
following Poincar\'e-type inequality holds.

\begin{lemma}[\cite{KLT09}] \label{L2.3}
 Let $\bar{u}\in P_{\lambda}G_{\tau}^{r+1}$
 and $\check{u}\in Q_\lambda G_{\tau}^{r+1}$. Then
\[
 |\overline{u}|_{r+1, \tau}
\leqslant e^{\tau \lambda^{1/2}}|\overline{u}|_{r+1}
 \quad \text{and}\quad
 |\check{u}|_{r, \tau}
\leqslant \lambda^{-1/2}|\check{u}|_{r+1,\tau}.
\]
\end{lemma}


\begin{lemma}[\cite{KLT09}] \label{L2.4}
 For any $\tau>0$, $u, w\in G_{\tau}^{2}$,
 and $v\in G_{\tau}^{1}$, the following inequality holds
\[
 \big|\big(B(u,v), w\big)_{1, \tau}\big|
\leqslant c(\Omega)\lambda_1^{-3/4}|u|^{1/2}_{1, \tau}|u|^{1/2}_{2, \tau}|v|_{1, \tau}|w|_{2, \tau}.
\]
\end{lemma}

\begin{lemma}[\cite{KLT09}] \label{L2.5}
 Let $s\in {\mathbb R}$. Assume that $h(t)\in L^\infty([0, T];
 V_{s-2})$ for some $T\in (0, \infty)$.
 Then the linear problem
\[
 z_t+\nu A z +\alpha^2 Az_t=h(t), \quad z(0)=0,
\]
has a unique solution $z(t)\in {\mathcal C}([0, T]; V_s)$. In
addition, the following estimate holds
\begin{align}\label{2.19}
 |z(t)|_s
\leqslant \frac{\|h(t)\|_{L^\infty([0, T];
 V_{s-2})}}{\alpha\nu\sqrt {(1/\lambda_1+\alpha^2)^{-1}}},
\end{align}
for all $t\in [0, T]$.
\end{lemma}

\begin{lemma}[\cite{KLT09}] \label{L2.6}
 Consider a nonnegative function $\psi(t)$ satisfying for all
 $t\geqslant t_0$, for some $t_0$, the following inequality
\[
 \frac{{\mathrm d} \psi}{{\mathrm d}t}
\leqslant
 -a\psi+d\psi^{3/2}+C, \quad \psi(t_0)=0,
\]
where the positive real coefficient $a, d, C$ obey
\begin{align}\label{2.20}
 dC^{1/2}< (a/2)^{3/2}.
\end{align}
Then for all $t\geqslant t_0$, there holds $\psi(t)\leqslant 2C/a$.
\end{lemma}

Finally, we will use the following lemma from Foias \emph{et al.}
\cite{FMRT04}.

\begin{lemma}[\cite{FMRT04}] \label{L2.7}
 Let $\psi(t)$ and $\phi(t)$ be locally integrable functions on
 $(0,+\infty)$ which satisfy some $T>0$ the conditions
\begin{gather*}
 \liminf_{t\to +\infty}\frac 1T\int_t^{t+T}\psi(\tau)
 {\mathrm d}\tau>0, \\
 \limsup_{t\to +\infty}\frac 1T\int_t^{t+T}\psi^{-}(\tau)
 {\mathrm d}\tau<\infty, \\
 \limsup_{t\to +\infty}\frac 1T\int_t^{t+T}\phi^{+}(\tau)
 {\mathrm d}\tau=0,
\end{gather*}
where $\psi^{-}=\max\{-\psi, 0\}$ and $\phi^{+}=\max\{\phi, 0\}$.
Suppose that $\varphi(t)$ is a nonnegative, absolutely continuous
function on $[0, +\infty)$ that satisfies the following inequality
a.e. on $[0, +\infty)$,
\[
 \varphi'(t)+\psi(t)\varphi(t)\leqslant \phi(t).
\]
 Then
$\varphi(t)\longrightarrow 0$ as $t\to +\infty$.
\end{lemma}


\section{Asymptotic approximation in $V_m$}

 The aim of this section is to construct an asymptotic
approximation of the solution of equations \eqref{2.14}-\eqref{2.16}
in the space $V_m$, for every $m\geqslant 2$. Note that
\[
\cap_{m=1}^\infty V_m \subset {\mathcal C}^\infty(\Omega).\]

\begin{theorem}\label{T3.1}
Let $m\geqslant 2$ be an integer, and let $(u(x,t), b(x,t))$ be a
solution of equations \eqref{2.14}-\eqref{2.16} corresponding to the
initial condition $(u^{\rm in}, b^{in})\in V\times V$ with the
forcing terms $(f, g)\in V_{m-2}\times V_{m-2}$. Then there exist
two functions
\[
 v^{(m)}(x,t)\in L^\infty ([0, +\infty);V_m)\quad
 \text{and}\quad \xi^{(m)}(x,t)\in L^\infty ([0, +\infty);V_m)
\]
satisfying
\[
 \lim_{t\to +\infty}|u(t)-v^{(m)}(t)|_1=
 \lim_{t\to +\infty}|b(t)-\xi^{(m)}(t)|_1=0.
\]
\end{theorem}

\begin{proof}
 Let us fixed $m\geqslant 2$, and let
$(u^{\rm in}, b^{\rm in})\in V\times V$. Firstly, we can write the solution as
 \begin{equation} \label{3.3}
 \begin{gathered}
 u(t)=v(t)+w(t) \\
 b(t)=\xi(t)+\eta(t),
 \end{gathered}
\end{equation}
where $(v(t), w(t))$ and $(\xi(t), \eta(t))$ satisfy the coupled equations
\begin{gather}
 v_t+\nu Av+\alpha^2A v_t=f-B(u, u)+B(b, b), \quad v(0)=0, \label{3.4}\\
 w_t+\nu A w+\alpha^2A w_t=0, \quad w(0)=u^{\rm in}, \label{3.5}
\end{gather}
and
\begin{gather}
 \xi_t+\mu A\xi+\beta^2A \xi_t=g-B(u, b)+B(b, u), \quad \xi(0)=0, \label{3.6}\\
\eta_t+\mu A \eta+\beta^2A \eta_t=0,
 \quad \eta(0)=b^{\rm in}, \label{3.7}
\end{gather}
respectively. By using  that both $u(x, t)$ and $b(x, t)$
belong to $L^\infty([0, +\infty); V)$, and applying Lemma
\ref{L2.1}(I) and Lemma \ref{L2.5}, we see that
\begin{equation} \label{3.8}
\begin{gathered}
 v(t)\in L^\infty([0, +\infty); V_{3/2})\\
 \xi(t)\in L^\infty([0, +\infty); V_{3/2}).
 \end{gathered}
\end{equation}
Now, by \eqref{3.5} and \eqref{3.7}, we conclude that
\begin{gather*}
 \|w(t)\|^2+\alpha^2|w|^2_1
\leqslant  e^{-\delta_1 t}(\|u^{\rm in}\|^2+\alpha^2|u^{\rm in}|^2_1), \\
 \|\eta(t)\|^2+\beta^2|\eta|^2_1
\leqslant  e^{-\delta_2 t}(\|b^{\rm in}\|^2+\beta^2|b^{\rm in}|^2_1),
 \end{gather*}
where[
$\delta_1:=(1/\lambda_1+\alpha^2)^{-1}$,
$\delta_2:=(1/\lambda_1+\beta^2)^{-1}$.
Therefore,
\begin{gather}
 \lim_{t\to+\infty}|u(t)-v(t)|_1
 =\lim_{t\to+\infty}|w(t)|_1=0, \label{3.S1}\\
 \lim_{t\to+\infty}|b(t)-\xi(t)|_1
 =\lim_{t\to+\infty}|\eta(t)|_1=0. \label{3.S2}
\end{gather}

At the next step, we consider $v^{(2)}(x, t)$, the solution of the equations
\begin{gather}
 v^{(2)}_t+ \nu A v^{(2)}+\alpha^2 A v^{(2)}_t=f-B(v,v)+B(\xi,\xi),
 \label{3.11} \\
 v^{(2)}(0)=0. \label{3.12}
\end{gather}
By Lemma \ref{L2.1} (II) and \eqref{3.8}, we see that the right-hand
side of equality \eqref{3.11} lies in $L^{\infty}([0, +\infty); H)$.
Hence, applying Lemma \ref{L2.5}, we find that the unique solution
of \eqref{3.11}-\eqref{3.12} satisfies
\begin{equation}\label{3.13}
 v^{(2)}(t)\in L^{\infty}([0, +\infty); V_2).
\end{equation}
Similarly, we consider $\xi^{(2)}(x, t)$, the solution of the equations
\begin{gather}
\xi^{(2)}_t+ \mu A \xi^{(2)}+\beta^2 A \xi^{(2)}_t=g-B(v,\xi)+B(\xi,v),
 \label{3.14} \\
\xi^{(2)}(0)=0. \label{3.15}
\end{gather}
We also can conclude that the unique solution of
\eqref{3.14}-\eqref{3.15} satisfies
\begin{equation} \label{3.13b}
 \xi^{(2)}(t)\in L^{\infty}([0, +\infty); V_2).
\end{equation}
Now, we set $z^{(2)}(x, t)=v^{(2)}(x, t)-v(x, t)$, which satisfies
\begin{gather}
\begin{aligned}
& z^{(2)}_t+\nu A z^{(2)}+\alpha^2 Az^{(2)}_t\\
&= B(u, u-v)+B(u-v,v) -B(b, b-\xi)-B(b-\xi,\xi),
\end{aligned} \label{3.17}\\
 z^{(2)}(0)= 0.\label{3.18}
\end{gather}
At the same time, we set $\eta^{(2)}(x, t)=\xi^{(2)}(x, t)-\xi(x,
t)$, which satisfies
\begin{gather}
\begin{aligned}
&\eta^{(2)}_t+\mu A \eta^{(2)}+\beta^2 A \eta^{(2)}_t\\
&=  B(u-v, \xi)+B(u,b-\xi)  -B(b-\xi, v)-B(b, u-v),
\end{aligned} \label{3.19}\\
 \eta^{(2)}(0)=  0.\label{3.20}
\end{gather}
Since $u, b\in L^\infty([0, +\infty); V)$, and $v$ and $\xi$ satisfy
\eqref{3.8}, we conclude from Lemma \ref{L2.5} that equations
\eqref{3.17}-\eqref{3.18} have a unique solution $z^{(2)}\in
L^\infty([0, +\infty); V_{3/2})$. Similarly, equations
\eqref{3.19}-\eqref{3.20} have a unique solution $\eta^{(2)}\in
L^\infty([0, +\infty); V_{3/2})$. Then taking an inner product of
\eqref{3.17} with $z^{(2)}$ and using \eqref{2.1}, we obtain
\begin{align*}
& \frac 12\frac{\mathrm{d}}{{\mathrm d}t}
 (\|z^{(2)}\|^2+\alpha^2|z^{(2)}|^2_1)
 +\nu|z^{(2)}|^2_1  \\
&=  B(u, u-v,z^{(2)})+B(u-v, v, z^{(2)})
 -B(b, b-\xi, z^{(2)})-B(b-\xi,\xi,z^{(2)}) \\
&\leqslant 
 c(\Omega)|u|_1\,|u-v|_1\, | z^{(2)}|_1
 +c(\Omega)|u-v|_1\,|v|_1\, |z^{(2)}|_1 \\
&\quad +c(\Omega)| b|_1\, |b-\xi|_1\, |z^{(2)}|_1
 +c(\Omega)|b-\xi|_1\, |\xi|_1\, |z^{(2)}|_1 \\
&\leqslant  c(\Omega,\nu)\big(|u-v|^2_1 (|u|^2_1+|v|^2_1)
 +|b-\xi|^2_1(|\xi|^2_1+|b|^2_1)\big)
 +\frac{\nu}{2}|z^{(2)}|^2_1,
\end{align*}
which gives
\begin{equation}
\begin{aligned}
&\frac{\mathrm{d}}{{\mathrm d}t}
 (\|z^{(2)}\|^2+\alpha^2|z^{(2)}|^2_1)
 +\frac{\nu \delta_1}{2}(\|z^{(2)}\|^2+\alpha^2|z^{(2)}|^2_1)  \\
&\leqslant   c(\Omega,\nu)\big(|u-v|^2_1 (|u|^2_1+|v|^2_1)
 +|b-\xi|^2_1(|\xi|^2_1+|b|^2_1)\big).
\end{aligned} \label{3.21}
\end{equation}
Analogous to the derivation of \eqref{3.21}, we can obtain
\begin{equation}
\begin{aligned}
&\frac{\mathrm{d}}{{\mathrm d}t}
 (\|\eta^{(2)}\|^2+\beta^2|\eta^{(2)}|^2_1)
 +\frac{\mu \delta_2}{2}(\|\eta^{(2)}\|^2+\beta^2|\eta^{(2)}|^2_1)  \\
&\leqslant
 c(\Omega,\nu)\big(|u-v|^2_1(|b|^2_1+|\xi|^2_1)
 +|b-\xi|^2_1 (|u|^2_1+|v|^2_1)\big).
 \label{3.22}
\end{aligned}
\end{equation}
By \eqref{3.S1}-\eqref{3.S2} and the fact that all
$u(t), v(t),b(t), \xi(t)$ are bounded uniformly in time in the $V$ norm, we
conclude from \eqref{3.21}-\eqref{3.22} and Lemma \ref{L2.7} that
\begin{gather*}
 \lim_{t\to+\infty}|z^{(2)}(t)|_1
 =\lim_{t\to+\infty}|v(t)-v^{(2)}(t)|_1=0, \\
 \lim_{t\to+\infty}|\eta^{(2)}(t)|_1
 =\lim_{t\to+\infty}|\xi(t)-\xi^{(2)}|_1=0.
\end{gather*}

We next continue by induction. Fix $2\leqslant n\leqslant m$, and
suppose that we have constructed $v^{(j)}(t)\in L^\infty([0,
+\infty); V_j)$ and $\xi^{(j)}(t)\in L^\infty([0, +\infty); V_j)$,
for $j=1,2, \cdots, n-1$, such that for any $j$
\begin{gather}
 \lim_{t\to+\infty}|v^{(j-1)}(t)-v^{(j)}(t)|_1
 =\lim_{t\to+\infty}|u(t)-v^{(j)}(t)|_1=0, \label{3.25}\\
\lim_{t\to+\infty}|\xi^{(j-1)}(t)-\xi^{(j)}(t)|_1
 =\lim_{t\to+\infty}|b(t)-\xi^{(j)}(t)|_1=0. \label{3.26}
\end{gather}
Let us consider the equations
\begin{gather}
 v^{(n)}_t+\nu A v^{(n)}+\alpha^2 A v^{(n)}_t
=f-B(v^{(n-1)}, v^{(n-1)})+B(\xi^{(n-1)}, \xi^{(n-1)}),
 \label{3.27} \\
 v^{(n)}(0)
=0, \label{3.28}
\end{gather}
and
\begin{gather}
 \xi^{(n)}_t+\mu A \xi^{(n)}+\beta^2 A \xi^{(n)}_t
=g-B(v^{(n-1)}, \xi^{(n-1)})+B(\xi^{(n-1)}, v^{(n-1)}),
 \label{3.29} \\
 \xi^{(n)}(0)
=0. \label{3.30}
\end{gather}
By Lemma \ref{L2.5} and the estimates on the nonlinear terms of
Lemma \ref{L2.1}, the unique solution $v^{(n)}(t)$ of
\eqref{3.27}-\eqref{3.28}, and $\xi^{(n)}(t)$ of
\eqref{3.29}-\eqref{3.30}, satisfy, respectively
\begin{gather*}
 v^{(n)}(t)\in L^\infty([0, +\infty); V_n), \\
 \xi^{(n)}(t)\in L^\infty([0, +\infty); V_n).
\end{gather*}
Furthermore, we denote $z^{(n)}(t)=v^{(n)}(t)-v^{(n-1)}(t)$, which
satisfies
\begin{gather}
\begin{aligned}
&z^{(n)}_t+\nu A z^{(n)}+\alpha^2 A z^{(n)}_t\\
&= B(v^{(n-2)}, v^{(n-2)}-v^{(n-1)})
 +B(v^{(n-2)}-v^{(n-1)}, v^{(n-1)})  \\
&\quad  -B(\xi^{(n-2)}, \xi^{(n-2)}-\xi^{(n-1)})
 -B(\xi^{(n-2)}-\xi^{(n-1)}, \xi^{(n-1)}),
\end{aligned}  \label{3.35} \\
 z^{(n)}(0) =0. \label{3.36}
\end{gather}
We also set
 $\eta^{(n)}(x,t)=\xi^{(n)}(x,t)-\xi^{(n-1)}(x,t)$, which satisfies
\begin{gather}
\begin{aligned}
&\eta^{(n)}_t+\mu A \eta^{(n)}+\beta^2 A \eta^{(n)}_t\\
&= B(v^{(n-2)}-v^{(n-1)}, \xi^{(n-1)})
 +B(v^{(n-2)}, \xi^{(n-2)}-\xi^{(n-1)})  \\
&\quad  -B(\xi^{(n-2)}-\xi^{(n-1)}, v^{(n-1)})
 -B(\xi^{(n-2)}, v^{(n-2)}-v^{(n-1)}),
\end{aligned} \label{3.37} \\
 z^{(n)}(0)=0. \label{3.38}
\end{gather}
Taking the inner product of equation \eqref{3.35} with $z^{(n)}(t)$
and using Lemma \ref{L2.1} and \eqref{3.25}-\eqref{3.26}, we get by
Lemma \ref{L2.7} that
\begin{equation} \label{3.39}
 \lim_{t\to+\infty}|z^{(n)}(t)|_1
= \lim_{t\to+\infty}|v^{(n-2)}(t)-v^{(n-1)}(t)|_1
= \lim_{t\to+\infty}|u(t)-v^{(n)}(t)|_1=0.
\end{equation}
Similarly, taking the inner product of equation \eqref{3.37} with
$\xi^{(n)}(t)$ and using Lemma \ref{L2.1} and
\eqref{3.25}-\eqref{3.26}, we get by Lemma \ref{L2.7} that
\begin{equation} \label{3.40}
 \lim_{t\to+\infty}|\eta^{(n)}(t)|_1
= \lim_{t\to+\infty}|\xi^{(n-2)}(t)-\xi^{(n-1)}(t)|_1
= \lim_{t\to+\infty}|b(t)-\xi^{(n)}(t)|_1=0.
\end{equation}
The proof of Theorem \ref{T3.1} is complete.
\end{proof}



\setcounter {equation}{0}
\section{Asymptotic approximation in the Gevrey space }

 The result of Section 3 shows that the global attractor of
equation \eqref{1.1}-\eqref{1.4} lies in ${\mathcal C}^\infty$
whenever the forcing terms $f$ and $g$ are ${\mathcal C}^\infty$.
The aim of this section is to show that the global attractor is real
analytic, whenever the forcing terms $f$ and $g$ are analytic. To
this end, we will borrow the idea of Oliver and Titi \cite{OT98,
OT00} and Kalantarov, Levant and Titi \cite{KLT09}, to construct the
asymptotic approximation of the solution of equations
\eqref{2.14}-\eqref{2.16} in the Gevrey space $G_\tau^2$, for some
$\tau>0$.

Firstly, it can be proved (see Catania\cite{C11}) that the solution
of the regularized MHD equations satisfies for all $t\geqslant 0$,
\begin{align*}
&\|u(t)\|^2+\alpha^2 |u(t)|^2_1+\|b(t)\|^2
 +\beta^2 |b(t)|^2_1  \\
&\leqslant
 e^{-\delta_5t}(\|u^{\rm in}\|^2
 +\alpha^2 |u^{\rm in}|^2_1
 +\|b^{\rm in}\|^2
 +\beta^2 |b^{\rm in}|^2_1-\frac{\delta_3}{\delta_5})
 +\frac{\delta_3}{\delta_5},
\end{align*}
where
\[
 \delta_3:= \frac{|f|^2_{-1}}{\nu}+\frac{|g|^2_{-1}}{\mu}, \quad
 \delta_4:= \min\{\mu, \nu\}, \quad
 \delta_5:= \frac{\delta_4}{2}\min\{\frac{1}{\alpha^2}, \frac{1}{\beta^2},
 \lambda_1\}.
\]
Therefore, there exists some $t_*$ depending on $\|u^{\rm in}\|$,
$|u^{\rm in}|_1$, $\|b^{\rm in}\|$, $|b^{\rm in}|_1$, $|f|_{-1}$,
$|g|_{-1}$, $\mu$, $\nu$, $\alpha$, $\beta$ and $\lambda_1$, such
that for all $t\geqslant t_*$
\begin{gather}
 |u(t)|_1\leqslant M_1/\alpha, \label{4.5} \\
 |b(t)|_1\leqslant M_1/\beta, \label{4.6}
\end{gather}
where $ M_1:=\sqrt{2\delta_3/\delta_5}$.
 Analogously, there exist two positive constants $M^{(1)}$ and
$M^{(2)}$, which depend only on $\|u^{\rm in}\|$, $|u^{\rm in}|_1$,
$\|b^{\rm in}\|$, $|b^{\rm in}|_1$, $|f|_{-1}$, $|g|_{-1}$, $\mu$,
$\nu$, $\alpha$, $\beta$ and $\lambda_1$, such that the solutions of
\eqref{3.11}-\eqref{3.12} and \eqref{3.14}-\eqref{3.15} satisfy
\begin{gather*}
 |v^{(2)}|_{1}
\leqslant  M^{(1)}_1, \quad \forall\, t\geqslant t_*, \\
|\xi^{(2)}|_{1} \leqslant  M^{(2)}_2, \quad \forall\, t\geqslant t_*.
\end{gather*}


\begin{lemma}\label{L4.1}
 Let $f$ and $g$ belong to $V_{m-2}$. Consider $t_*\geqslant 0$,
such that the solution of the regularized MHD equations
\eqref{2.14}-\eqref{2.16} satisfies the relations \eqref{4.5} and
\eqref{4.6} for all $t\geqslant t_*$. Then the following statements
are true
\begin{itemize}
\item[(1)] The functions $v(x,t)$, $\xi(x,t)\in L^\infty([0, +\infty);
V_{3/2})$, constructed in Theorem \ref{T3.1} satisfy for all
$t\geqslant t_*$,
\begin{gather*}
 |v(t)|_{3/2}
\leqslant  M^{(1)}_{3/2}
:=  \frac{|f|_{-1/2}+c(\Omega)\lambda_1^{-\frac 34}M_1^2
 (\frac{1}{\alpha^2}+\frac{1}{\beta^2})}
{\alpha\nu\sqrt{(\lambda_1^{-1}+\alpha^2)^{-1}}} , \\
|\xi(t)|_{3/2}
\leqslant  M^{(2)}_{3/2}
:=  \frac{|g|_{-1/2}+c(\Omega)\lambda_1^{-\frac 34}M_1^2
(\frac{1}{\alpha^2}+\frac{1}{\beta^2})}
 {\beta\mu\sqrt{(\lambda_1^{-1}+\beta^2)^{-1}}} .
\end{gather*}

\item[(2)] The functions $v^{(2)}(x,t)$,
$\xi^{(2)}(x,t)\in L^\infty([0, +\infty); V_{2})$, constructed in Theorem
\ref{T3.1} satisfy for all $t\geqslant t_*$,
\begin{gather*}
 |v^{(2)}(t)|_{2}
\leqslant M^{(1)}_{2}
:= \frac{|f|+c(\Omega)\lambda_1^{-\frac 34}M_1
 (\frac{M^{(1)}_{3/2}}{\alpha}+\frac{M^{(2)}_{3/2}}{\beta})}
 {\alpha\nu\sqrt{(\lambda_1^{-1}+\alpha^2)^{-1}}},\\
 |\xi^{(2)}(t)|_{2}
\leqslant  M_{2}^{(2)}
:= \frac{|g|+c(\Omega)\lambda_1^{-\frac 34}M_1
 (\frac{M^{(2)}_{3/2}}{\alpha}+\frac{M^{(1)}_{3/2}}{\beta})}
 {\beta\mu\sqrt{(\lambda_1^{-1}+\beta^2)^{-1}}}.
\end{gather*}

\item[(3)] Let $m$ be an integer. The functions
$v^{(m)}(x,t)$, $\xi^{(m)}(x,t)\in L^\infty([0, +\infty);V_{m})$,
 constructed in Theorem \ref{T3.1} satisfy for all $t\geqslant t_*$,
\[
 |v^{(m)}(t)|_{m}
\leqslant  M^{(1)}_{m}, \quad
 |\xi^{(m)}(t)|_{m}
\leqslant  M^{(2)}_{m},
\]
where
\begin{gather*}
 M^{(1)}_{m}
:= \frac{|f|_{m-2}+c(m,\Omega)\lambda_1^{-\frac 78}
 \big[
 (\frac{M_1}{\alpha})^{\frac 14}
 (M^{(1)}_{2})^{\frac 34}M^{(1)}_{m-1}
 +(\frac{M_1}{\beta})^{\frac 14}
 (M^{(2)}_{2})^{\frac 34}M^{(2)}_{m-1}\big]}
 {\alpha\nu\sqrt{(\lambda_1^{-1}+\alpha^2)^{-1}}},
\\
 M^{(2)}_{m}
:= \frac{|f|_{m-2}+c(m,\Omega)\lambda_1^{-\frac 78}
 \big[
 (\frac{M_1}{\alpha})^{\frac 14}
 (M^{(2)}_{2})^{\frac 34}M^{(1)}_{m-1}
 +(\frac{M_1}{\beta})^{\frac 14}
 (M^{(2)}_{2})^{\frac 34}M^{(1)}_{m-1}\big]}
 {\beta\mu\sqrt{(\lambda_1^{-1}+\beta^2)^{-1}}}.
\end{gather*}
\end{itemize}
\end{lemma}


\begin{proof}
 Recall that $v(x,t)$ and $\xi(x,t)$ satisfy \eqref{3.4} and
 \eqref{3.6}, respectively. In general, $v^{(m)}(x,t)$ and $\xi^{(m)}(x,t)$,
for $m>2$, satisfy \eqref{3.27} and
 \eqref{3.29}, respectively. Therefore, the proof of this lemma is
 an immediate application of Lemma \ref{L2.5}, in particular
 relation \eqref{2.19}, and the inequalities of Lemma \ref{L2.1}.
\end{proof}


\begin{theorem}\label{T4.1}
Let $(u(x, t), b(x, t))$ be a solution of the regularized MHD
equations \eqref{2.14}-\eqref{2.16}, corresponding to the initial
condition $(u^{\rm in}, b^{\rm in})\in V\times V$ with forcing terms
$(f, g)\in G_{\tau_0}^1\times G_{\tau_0}^1$, for some $\tau_0>0$.
Let $t_*\geqslant 0$ be as in Lemma \ref{L4.1}, then there exist two
functions
\begin{gather}
 v^{\omega}(t)\in L^\infty([t_*, +\infty); G_\tau^2), \label{4.14}\\
 \xi^{\omega}(t)\in L^\infty([t_*, +\infty); G_\tau^2), \label{4.15}
\end{gather}
for some $\tau>0$, depending only on $|f|_{1, \tau_0}$,
$|g|_{1,\tau_0}$, $\mu$, $\nu$, $\alpha$, $\beta$ and $\lambda_1$,
satisfying
\begin{gather}
 \lim_{t\to +\infty}|u(t)-v^{\omega}(t)|_1=0, \label{4.16}\\
 \lim_{t\to +\infty}|b(t)-\xi^{\omega}(t)|_1=0. \label{4.17}
\end{gather}
\end{theorem}

\begin{proof}
 Let $\lambda>0$ to be chosen later. First, consider
 $(v^{(2)}(x, t), \xi^{(2)}(x, t))$ the asymptotic approximation of
 $(u(x, t), b(x, t))$, which is constructed in the proof of Theorem
\ref{T3.1}.
Denote
\[
 \overline{v}(t)=P_\lambda v^{(2)}(t), \quad
\overline{\xi}(t)=P_\lambda \xi^{(2)}(t).
\]
Consider $\check{v}(t)$ and $\check{\xi}(t)$ for all $t\geqslant t_*$
-the solution of the following equations
\begin{gather}
\check{v}_t
 +\nu A \check{v}
 +\alpha^2A \check{v}_t
 +Q_\lambda B(\overline{v}+\check{v}, \overline{v}+\check{v})
 -Q_\lambda B(\overline{\xi}+\check{\xi}, \overline{\xi}+\check{\xi})
 = \check{f}, \label{4.20}\\
\check{v}(t_*)=0, \label{4.21}
\end{gather}
and
\begin{gather}
 \check{\xi}_t
 +\mu A \check{\xi}
 +\beta^2A \check{\xi}_t
 +Q_\lambda B(\overline{v}+\check{v}, \overline{\xi}+\check{\xi})
 -Q_\lambda B(\overline{\xi}+\check{\xi}, \overline{v}+\check{v})
 = \check{g}, \label{4.22}\\
 \check{\xi}(t_*)=0, \label{4.23}
\end{gather}
respectively, where $\check{f}=Q_\lambda f$ and $\check{g}=Q_\lambda g$. Let us put
\begin{gather}
 v^{\omega}(t)=\overline{v}(t)+\check{v}(t), \quad t\geqslant
 t_*,\label{4.24}\\
 \xi^{\omega}(t)=\overline{\xi}(t)+\check{\xi}(t), \quad t\geqslant
 t_*.\label{4.25}
\end{gather}
Our goal is to show first that there exists some $\tau>0$ such that
\begin{align*}
 v^{\omega}(t),\quad
 \xi^{\omega}(t)\in G_\tau^2.
\end{align*}
Since $\overline{v}(t)$ and $\overline{\xi}(t)$ are just
trigonometric polynomials, and in particular, are analytic, we need
to show that we can choose $\lambda$ large enough, such that
$\check{v}(t)$ and $\check{\xi}(t)$ belong to $G_\tau^2$ for some
$\tau>0$. Finally, we will show that $(v^{\omega}(x, t),
\xi^{\omega}(x, t))$ is indeed the asymptotic approximation of
$(u(x, t), b(x, t))$.

We want to point out that in order to prove that the solutions of
\eqref{4.20}-\eqref{4.21} and \eqref{4.22}-\eqref{4.23} belong to a
Gevrey class of real analytic functions, we consider the Galerkin
procedure to \eqref{4.20} and \eqref{4.22}, respectively. However,
we omit this standard procedure, and obtain formal a priori
estimates on the solutions in relevant Gevrey space norm. Taking
formally the inner product of \eqref{4.20} with $\check{v}$, and
\eqref{4.22} with $\check{\xi}$ in $G_\tau^1$, respectively, we
obtain the following inequalities
\begin{equation} \label{4.28}
\begin{aligned}
&\frac12(|\check{v}|^2_{1, \tau}+\alpha^2|\check{v}|^2_{2, \tau})
 +\nu|\check{v}|^2_{2, \tau}\\
&\leqslant  |(\check{f}, \check{v})_{1,\tau}|
 +|(B(\overline{v} ,\overline{v}), \check{v})_{1,\tau}|
+|(B(\overline{v}, \check{v}), \check{v})_{1,\tau}|
 +|(B(\check{v} ,\overline{v}), \check{v})_{1,\tau}| \\
&\quad  +|(B(\check{v}, \check{v}), \check{v})_{1,\tau}|
 +|(B(\overline{\xi}, \overline{\xi}), \check{v})_{1,\tau}|
 +|(B(\overline{\xi}, \check{\xi}), \check{v})_{1,\tau}| \\
&\quad +|(B(\check{\xi}, \overline{\xi}), \check{v})_{1,\tau}| 
  +|(B(\check{\xi}, \check{\xi}), \check{v})_{1,\tau}|
\end{aligned}
\end{equation}
and
\begin{equation} \label{4.29}
\begin{aligned}
&\frac12(|\check{\xi}|^2_{1, \tau}+\beta^2|\check{\xi}|^2_{2, \tau})
 +\mu|\check{\xi}|^2_{2, \tau}\\
&\leqslant
 |(\check{g}, \check{\xi})_{1,\tau}|
 +|(B(\overline{v} ,\overline{\xi}), \check{\xi})_{1,\tau}|
 +|(B(\overline{v}, \check{\xi}), \check{\xi})_{1,\tau}|
 +|(B(\check{v} ,\overline{\xi}), \check{\xi})_{1,\tau}| \\
&\quad +|(B(\check{v}, \check{\xi}), \check{\xi})_{1,\tau}|
 +|(B(\overline{\xi}, \overline{v}), \check{\xi})_{1,\tau}|
 +|(B(\overline{\xi}, \check{v}), \check{\xi})_{1,\tau}|\\
&\quad +|(B(\check{\xi}, \overline{v}), \check{\xi})_{1,\tau}| 
  +|(B(\check{\xi}, \check{v}), \check{v})_{1,\tau}|
\end{aligned}
\end{equation}
accordingly. We next estimate the terms on the right-hand side of
above \eqref{4.28} and \eqref{4.29}.


Firstly, using subsequently the Cauchy-Schwarz and Young
inequalities, as well as Lemma \ref{L2.3}, we obtain assuming
$\tau\leqslant \tau_0$ that
\begin{gather}
 |(\check{f}, \check{v})_{1,\tau}|
\leqslant
 |\check{f}|_{1,\tau}|\check{v}|_{1,\tau}
\leqslant
 \frac{2}{\nu\lambda}|\check{f}|^2_{1,\tau}
 +\frac{\nu}{8}|\check{v}|^2_{2,\tau}, \label{4.30}\\
 |(\check{g}, \check{\xi})_{1,\tau}|
\leqslant
 |\check{g}|_{1,\tau}|\check{\xi}|_{1,\tau}
\leqslant
 \frac{2}{\mu\lambda}|\check{g}|_{1,\tau}^2
 +\frac{\mu}{8}|\check{\xi}|^2_{2,\tau}. \label{4.31}
\end{gather}
Secondly, using Lemma \ref{L2.4}, Young inequality and the
Poincar\'e-type inequalities of Lemma \ref{L2.3}, we obtain the
following series of estimates for all $t\geqslant t_*$:
\begin{gather}
\begin{aligned}
 |(B(\overline{v} ,\overline{v}), \check{v})_{1,\tau}|
&\leqslant
 c(\Omega)\lambda_1^{-3/4}|\overline{v}|^{3/2}_{1,\tau}
 |\overline{v}|^{1/2}_{2,\tau}|\check{v}|_{2,\tau}  \\
&\leqslant
 \frac{c(\Omega)e^{4\tau\lambda^{1/2}}(M_1^{(1)})^3M_2^{(1)}}{\nu\lambda_1^{3/2}}
 +\frac{\nu}{8}|\check{v}|^2_{2,\tau},
\end{aligned}\label{4.32}\\
\begin{aligned}
 |(B(\overline{v}, \check{v}), \check{v})_{1,\tau}|
&\leqslant
 c(\Omega)\lambda_1^{-3/4}|\overline{v}|^{1/2}_{1,\tau}
 |\overline{v}|^{1/2}_{2,\tau}
 |\check{v}|_{1,\tau}|\check{v}|_{2,\tau}  \\
&\leqslant
 \frac{c(\Omega)e^{\tau \lambda^{1/2}}
 (M_1^{(1)}M_2^{(1)})^{1/2}}{\lambda^{1/2}\lambda_1^{3/4}}
 |\check{v}|^2_{2,\tau},
\end{aligned} \label{4.33}\\
 |(B(\check{v}, \overline{v}), \check{v})_{1,\tau}|
\leqslant
 c(\Omega)\lambda_1^{-3/4}|\check{v}|^{1/2}_{1,\tau}
 |\check{v}|^{3/2}_{2,\tau}
 |\overline{v}|_{1,\tau}
\leqslant  \frac{c(\Omega)e^{\tau \lambda^{1/2}}M^{(1)}_1}{\lambda^{1/4}\lambda_1^{3/4}}
 |\check{v}|^2_{2,\tau},
\label{4.34}\\
\begin{aligned}
 |(B(\check{v}, \check{v}), \check{v})_{1,\tau}|
&\leqslant  c(\Omega)\lambda_1^{-3/4}|\check{v}|^{3/2}_{1,\tau}
 |\check{v}|^{3/2}_{2,\tau}
\leqslant  c(\Omega)\lambda^{-3/4}\lambda_1^{-3/4} |\check{v}|^{3}_{2,\tau}
  \\
&\leqslant  \frac{c(\Omega)}{\lambda^{3/4}\lambda_1^{3/4}\alpha^{3}}
 (|\check{v}|^2_{1,\tau}+\alpha^2|\check{v}|^2_{2,\tau})^{3/2},
\end{aligned} \label{4.35}\\
\begin{aligned}
 |(B(\overline{\xi} ,\overline{\xi}), \check{v})_{1,\tau}|
&\leqslant
 c(\Omega)\lambda_1^{-3/4}|\overline{\xi}|^{3/2}_{1,\tau}
 |\overline{\xi}|^{1/2}_{2,\tau}|\check{v}|_{2,\tau}  \\
&\leqslant
 \frac{c(\Omega)e^{4\tau\lambda^{1/2}}(M^{(2)}_1)^3M^{(2)}_2}{\nu\lambda_1^{3/2}}
 +\frac{\nu}{8}|\check{v}|^2_{2,\tau},
\end{aligned}\label{4.36}\\
\begin{aligned}
 |(B(\overline{\xi}, \check{\xi}), \check{v})_{1,\tau}|
&\leqslant  c(\Omega)\lambda_1^{-4/3}|\overline{\xi}|^{1/2}_{1,\tau}
 |\overline{\xi}|^{1/2}_{2,\tau}
 |\check{\xi}|_{1,\tau}
 |\check{v}|_{2,\tau}  \\
&\leqslant
 \frac{c(\Omega)e^{\tau \lambda^{1/2}}(M^{(2)}_1M^{(2)}_2)^{1/2}}
 {\lambda^{1/2}\lambda_1^{3/4}}
 (|\check{v}|^2_{2,\tau}+|\check{\xi}|^2_{2,\tau}),
\end{aligned} \label{4.37}\\
\begin{aligned}
 |(B(\check{\xi}, \overline{\xi}), \check{v})_{1,\tau}|
&\leqslant
 c(\Omega)\lambda_1^{-3/4}|\check{\xi}|^{1/2}_{1,\tau}
 |\check{\xi}|^{1/2}_{2,\tau}
 |\overline{\xi}|_{1,\tau}
 |\check{v}|_{2,\tau} \\
&\leqslant
 c(\Omega)\lambda_1^{-3/4}\lambda^{-1/4}
 |\check{\xi}|_{2,\tau}
 |\overline{\xi}|_{1,\tau}
 |\check{v}|_{2,\tau} \\
&\leqslant
 \frac{c(\Omega)e^{\tau \lambda^{1/2}}M^{(2)}_1}{\lambda_1^{3/4}\lambda^{1/4}}
 (|\check{v}|^2_{2,\tau}+|\check{\xi}|^2_{2,\tau}),
\end{aligned} \label{4.38}\\
\begin{aligned}
& |(B(\check{\xi}, \check{\xi}), \check{v})_{1,\tau}|\\
&\leqslant  c(\Omega)\lambda_1^{-3/4}|\check{\xi}|^{3/2}_{1,\tau}
 |\check{\xi}|^{1/2}_{2,\tau}
 |\check{v}|_{2,\tau}
\leqslant  c(\Omega)\lambda^{-3/4}\lambda_1^{-3/4}
 |\check{\xi}|^{2}_{2,\tau}|\check{v}|_{2,\tau}  \\
&\leqslant  c(\Omega)\lambda^{-3/4}\lambda_1^{-3/4}\alpha^{-3}
 (|\check{v}|^2_{1,\tau}+\alpha^2|\check{v}|^2_{2,\tau})^{3/2} \\
&\quad
 +c(\Omega)\lambda^{-3/4}\lambda_1^{-3/4}\beta^{-3}
 (|\check{\xi}|^2_{1,\tau}+\beta^2|\check{\xi}|^2_{2,\tau})^{3/2} \\
&\leqslant
 c(\Omega,\alpha,\beta)\lambda^{-3/4}\lambda_1^{-3/4}
 (|\check{v}|^2_{1,\tau}+\alpha^2|\check{v}|^2_{2,\tau}
 +|\check{\xi}|^2_{1,\tau}+\beta^2|\check{\xi}|^2_{2,\tau})^{3/2},
\end{aligned} \label{4.39}\\
\begin{aligned}
 |(B(\overline{v} ,\overline{\xi}), \check{\xi})_{1,\tau}|
&\leqslant
 c(\Omega)\lambda_1^{-3/4}
 |\overline{v}|^{1/2}_{1,\tau}
 |\overline{v}|^{1/2}_{2,\tau}
 |\overline{\xi}|_{1,\tau}|\check{\xi}|_{2,\tau}  \\
&\leqslant
 \frac{c(\Omega)e^{4\tau\lambda^{1/2}}M^{(1)}_1M^{(1)}_2(M_1^{(2)})^2}{\mu\lambda_1^{3/2}}
 +\frac{\mu}{8}|\check{\xi}|^2_{2,\tau},
\end{aligned}\label{4.40}\\
\begin{aligned}
 |(B(\overline{v}, \check{\xi}), \check{\xi})_{1,\tau}|
&\leqslant
 c(\Omega)\lambda_1^{-3/4}
 |\overline{v}|^{1/2}_{1,\tau}
 |\overline{v}|^{1/2}_{2,\tau}
 |\check{\xi}|_{1,\tau}
 |\check{\xi}|_{2,\tau}  \\
&\leqslant
 \frac{c(\Omega)
 e^{\tau \lambda^{1/2}}(M^{(1)}_1M^{(1)}_2)^{1/2}}{\lambda^{1/2}\lambda_1^{3/4}}
 |\check{\xi}|^2_{2,\tau},
\end{aligned} \label{4.41} \\
\begin{aligned}
 |(B(\check{v}, \overline{\xi}), \check{\xi})_{1,\tau}|
&\leqslant
 c(\Omega)\lambda_1^{-3/4}
 |\check{v}|^{1/2}_{1,\tau}
 |\check{v}|^{1/2}_{2,\tau}
 |\overline{\xi}|_{1,\tau}
 |\check{\xi}|_{2,\tau}  \\
&\leqslant
 c(\Omega)\lambda_1^{-3/4}\lambda^{-1/4}M^{(2)}_1e^{\tau\lambda^{1/2}}
 (|\check{v}|^2_{2,\tau}
 + |\check{\xi}|^2_{2,\tau}),
\end{aligned} \label{4.42}\\
\begin{aligned}
&|(B(\check{v}, \check{\xi}), \check{\xi})_{1,\tau}|\\
&\leqslant
 c(\Omega)\lambda_1^{-3/4}|\check{v}|^{1/2}_{1,\tau}
 |\check{v}|^{1/2}_{2,\tau}
 |\check{\xi}|_{1,\tau}
 |\check{\xi}|_{2,\tau}\\
&\leqslant
 c(\Omega)\lambda^{-3/4}\lambda_1^{-3/4}
 |\check{v}|_{2,\tau}
 |\check{\xi}|^{2}_{2,\tau}
  \\
&\leqslant
 c(\Omega,\alpha,\beta)\lambda^{-3/4}\lambda_1^{-3/4}
 (|\check{v}|^2_{1,\tau}+\alpha^2|\check{v}|^2_{2,\tau}
 +|\check{\xi}|^2_{1,\tau}+\beta^2|\check{\xi}|^2_{2,\tau})^{3/2},
\end{aligned}\label{4.43}\\
\begin{aligned}
 |(B(\overline{\xi}, \overline{v}), \check{\xi})_{1,\tau}|
&\leqslant
 c(\Omega)\lambda_1^{-3/4}
 |\overline{\xi}|^{1/2}_{1,\tau}
 |\overline{\xi}|^{1/2}_{2,\tau}
 |\overline{v}|_{1,\tau}
 |\check{\xi}|_{2,\tau}  \\
&\leqslant
 \frac{c(\Omega)e^{4\tau \lambda^{1/2}}M^{(2)}_1M^{(2)}_2(M^{(1)}_1)^2}{\mu\lambda_1^{3/2}}
 +\frac{\mu}{8}|\check{\xi}|^2_{2,\tau},
\end{aligned} \label{4.44}\\
\begin{aligned}
 |(B(\overline{\xi}, \check{v}), \check{\xi})_{1,\tau}|
&\leqslant
 c(\Omega)\lambda_1^{-3/4}
 |\overline{\xi}|^{1/2}_{1,\tau}
 |\overline{\xi}|^{1/2}_{2,\tau}
 |\check{v}|_{1,\tau}
 |\check{\xi}|_{2,\tau}  \\
&\leqslant
 \frac{ce^{\tau \lambda^{1/2}}(M_1^{(2)}M_2^{(2)})^{1/2}}{\lambda^{1/2}\lambda_1^{3/4}}
 (|\check{v}|^2_{2,\tau}+|\check{\xi}|^2_{2,\tau}),
\end{aligned}\label{4.45}\\
\begin{aligned}
 |(B(\check{\xi}, \overline{v}), \check{\xi})_{1,\tau}|
&\leqslant
 c(\Omega)\lambda_1^{-3/4}
 |\check{\xi}|^{1/2}_{1,\tau}
 |\check{\xi}|^{1/2}_{2,\tau}
 |\overline{v}|_{1,\tau}
 |\check{\xi}|_{2,\tau} \\
&\leqslant
 \frac{c(\Omega)e^{\tau \lambda^{1/2}}M^{(1)}_1}{\lambda^{1/4}\lambda_1^{3/4}}
 |\check{\xi}|^2_{2,\tau},
\end{aligned} \label{4.46}\\
\begin{aligned}
& |(B(\check{\xi}, \check{v}), \check{\xi})_{1,\tau}|\\
&\leqslant
 c(\Omega)\lambda_1^{-3/4}
 |\check{\xi}|^{1/2}_{1,\tau}
 |\check{\xi}|^{1/2}_{2,\tau}
 |\check{v}|_{1,\tau}
 |\check{\xi}|_{2,\tau} \\
&\leqslant
 c\lambda^{-1/4}\lambda_1^{-3/4} |\check{\xi}|^{2}_{2,\tau}|\check{v}|_{2,\tau}
  \\
&\leqslant
 c(\Omega,\alpha,\beta)\lambda^{-3/4}\lambda_1^{-3/4}
 (|\check{v}|^2_{1,\tau}+\alpha^2|\check{v}|^2_{2,\tau}
 +|\check{\xi}|^2_{1,\tau}+\beta^2|\check{\xi}|^2_{2,\tau})^{3/2},\label{4.47}
\end{aligned}
\end{gather}
Let us set $\tau :=\min\{\lambda^{-1/2}, \tau_0\}$. Then we select
$\lambda$ large enough satisfying
\begin{equation} \label{4.48}
\begin{aligned}
&\max\Big \{\frac{c(\Omega)
 [(M_1^{(1)}M_2^{(1)})^{1/2}+M^{(2)}_1M^{(2)}_2)^{1/2}]}{\lambda^{1/2}\lambda_1^{3/4}},
 \frac{c(\Omega)(M^{(1)}_1+M^{(2)}_1)}{\lambda^{1/4}\lambda_1^{3/4}}
 \Big\} \\
&\leqslant \frac{\min\{\nu, \mu\}}{8}.
\end{aligned}
\end{equation}
Taking \eqref{4.28}-\eqref{4.48} into account, we obtain
\begin{equation} \label{4.49}
\begin{aligned}
& \frac12
 \frac{{\mathrm d}}{{\mathrm d}t}
 (|\check{v}|^2_{1, \tau}+\alpha^2|\check{v}|^2_{2, \tau}
 +|\check{\xi}|^2_{1, \tau}+\beta^2|\check{\xi}|^2_{2, \tau}
 )
 +\frac{\nu}{8}|\check{v}|^2_{2, \tau}
 +\frac{\mu}{8}|\check{\xi}|^2_{2, \tau}  \\
&\leqslant
 \frac{c(\Omega,\alpha,\beta)}{\lambda^{3/4}\lambda_1^{3/4}}
 (1+\frac{1}{\alpha^3})
 (|\check{v}|^2_{1,\tau}+\alpha^2|\check{v}|^2_{2,\tau}
 +|\check{\xi}|^2_{1,\tau}+\beta^2|\check{\xi}|^2_{2,\tau})^{3/2}  \\
&\quad +\frac{2}{\nu\lambda}|\check{f}|^2_{1,\tau}
 +\frac{2}{\mu\lambda}|\check{g}|^2_{1,\tau}
 +\frac{c(\Omega)(M_1^{(1)})^3M_2^{(1)}}{\nu\lambda_1^{3/2}}
 +\frac{c(\Omega)(M^{(2)}_1)^3M^{(2)}_2}{\nu\lambda_1^{3/2}}  \\
&\quad  +\frac{c(\Omega)M^{(1)}_1M^{(1)}_2(M_1^{(2)})^2}{\mu\lambda_1^{3/2}}
 +\frac{c(\Omega)M^{(2)}_1M^{(2)}_2(M^{(1)}_1)^2}{\mu\lambda_1^{3/2}}.
\end{aligned}
\end{equation}
Using Lemma \ref{L2.3} and setting
\begin{align*}
 \delta_6
 :=\min\{\delta_1, \delta_2\}
 =\min\{(1/\lambda_1+\alpha^2)^{-1}, (1/\lambda_1+\beta^2)^{-1}\},
\end{align*}
we can write
\begin{gather}
 \frac{\nu}{8}|\check{v}|^2_{2, \tau}
\geqslant
 \frac{\nu \delta_6}{8}
 (\lambda^{-1}|\check{v}|^2_{2, \tau}
 +\alpha^2|\check{v}|^2_{2, \tau})
\geqslant
 \frac{\nu \delta_6}{8}
 (|\check{v}|^2_{1, \tau}
 +\alpha^2|\check{v}|^2_{2, \tau}), \label{4.51}\\
 \frac{\mu}{8}|\check{\xi}|^2_{2, \tau}
\geqslant
 \frac{\mu \delta_6}{8}
 (\lambda^{-1}|\check{\xi}|^2_{2, \tau}
 +\beta^2|\check{\xi}|^2_{2, \tau})
\geqslant
 \frac{\mu \delta_6}{8}
 (|\check{\xi}|^2_{1, \tau}
 +\beta^2|\check{\xi}|^2_{2, \tau}). \label{4.52}
\end{gather}
Substituting \eqref{4.51}-\eqref{4.52} into \eqref{4.49} gives
\begin{equation}  \label{4.54}
\begin{aligned}
& \frac12 \frac{{\mathrm d}}{{\mathrm d}t}
 (|\check{v}|^2_{1, \tau}+\alpha^2|\check{v}|^2_{2, \tau}
 +|\check{\xi}|^2_{1, \tau}+\beta^2|\check{\xi}|^2_{2, \tau} )  \\
&\leqslant
 -\frac{\delta_4\delta_6}{8}(|\check{v}|^2_{1,\tau}+\alpha^2|\check{v}|^2_{2,\tau}
 +|\check{\xi}|^2_{1,\tau}+\beta^2|\check{\xi}|^2_{2,\tau})
  \\
&\quad  + \frac{c(\Omega,\alpha,\beta)}{\lambda^{3/4}\lambda_1^{3/4}}
 (1+\frac{1}{\alpha^3})
 (|\check{v}|^2_{1,\tau}+\alpha^2|\check{v}|^2_{2,\tau}
 +|\check{\xi}|^2_{1,\tau}+\beta^2|\check{\xi}|^2_{2,\tau})^{3/2}
  \\
&\quad  +\frac{2}{\nu\lambda}|\check{f}|^2_{1,\tau}
 +\frac{2}{\mu\lambda}|\check{g}|^2_{1,\tau}
 +K,
\end{aligned}
\end{equation}
where the constant
\begin{align*}
 K&:=\frac{c(\Omega)}{\lambda_1^{3/2}}\Big[
 \frac{(M_1^{(1)})^3M_2^{(1)}+(M^{(2)}_1)^3M^{(2)}_2}{\nu}\\
 &\quad +\frac{M^{(1)}_1M^{(1)}_2(M_1^{(2)})^2
 +M^{(2)}_1M^{(2)}_2(M^{(1)}_1)^2}{\mu}\Big]
\end{align*}
is independent of $\lambda$.


To apply Lemma \ref{L2.6} to the function
$(|\check{v}|^2_{1, \tau}+\alpha^2|\check{v}|^2_{2, \tau}
 +|\check{\xi}|^2_{1, \tau}+\beta^2|\check{\xi}|^2_{2, \tau} )$ satisfying
inequality \eqref{4.54}, we need check the condition \eqref{2.20}.
In fact, we can choose $\lambda$ large enough, such that
\begin{equation} \label{4.56}
\frac{c(\Omega,\alpha,\beta)(1+\frac{1}{\alpha^3})}{\lambda^{3/4}\lambda_1^{3/4}}
 \Big(\frac{2(\nu|\check{f}|^2_{1,\tau}
 +\mu|\check{g}|^2_{1,\tau})}{\lambda}
 +K\Big)^{1/2}
< (\frac{\delta_4\delta_6}{16})^{3/2}.
\end{equation}
For such choice of $\lambda$, we conclude from Lemma \ref{L2.5} that
$(|\check{v}|^2_{1, \tau}+\alpha^2|\check{v}|^2_{2, \tau}
 +|\check{\xi}|^2_{1, \tau}+\beta^2|\check{\xi}|^2_{2, \tau}
)$ is bounded for all $t\geqslant t_*$, and hence
\[
 v^{\omega}(t)\in L^\infty([t_*; +\infty); G_\tau^2)\quad\text{and}\quad
 \xi^{\omega}(t)\in L^\infty([t_*; +\infty); G_\tau^2),
\]
that is \eqref{4.14} and \eqref{4.15} are proved.

We now are left to show that $(v^{\omega}(x,t), \xi^{\omega}(x,t))$
is the asymptotic approximation of the solution $(u(x,t), b(x,t))$
of the regularized MHD equations. Put
\begin{gather*}
 z=u-v^\omega,\quad \overline{z}
= P_\lambda(u-v^{(2)})
= P_\lambda u-\overline{v}, \\
 \zeta=b-\xi^\omega, \quad \overline{\zeta}
= P_\lambda(b-\xi^{(2)})
= P_\lambda b-\overline{\xi},\quad
 \end{gather*}
and denote
\begin{gather}
 \check{z}= Q_\lambda u-\check{v}, \label{4.61}\\
 \check{\zeta}= Q_\lambda b-\check{\xi}. \label{4.62}
\end{gather}
Relations \eqref{4.24}-\eqref{4.25} and \eqref{4.61}-\eqref{4.62}
give
\begin{equation}  \label{4.S3}
\begin{gathered}
 \overline{z}+\check{z}=u-v^\omega=z, \\
 \overline{\zeta}+\check{\zeta}=b-\xi^\omega=\zeta.
 \end{gathered}
\end{equation}
Obviously, by the construction and Theorem \ref{T3.1}, we have
\begin{gather}
 \lim_{t\to +\infty}
 |P_\lambda u(t)-\overline{v}(t)|_1
= \lim_{t\to +\infty}|\overline{z}|_1=0,
 \label{4.63}\\
 \lim_{t\to +\infty}
 |P_\lambda b(t)-\overline{\xi}(t)|_1
= \lim_{t\to +\infty}|\overline{\zeta}|_1=0. \label{4.64}
\end{gather}
Therefore, to prove \eqref{4.16} and \eqref{4.17}, we need to show
\begin{gather}
 \lim_{t\to +\infty} |Q_\lambda u(t)-\check{v}(t)|_1
= \lim_{t\to +\infty}|\check{z}(t)|_1=0, \label{4.65}\\
 \lim_{t\to +\infty}
 |Q_\lambda b(t)-\check{\xi}(t)|_1
= \lim_{t\to +\infty}|\check{\zeta}(t)|_1=0. \label{4.66}
\end{gather}
By the property of the operator $B(\cdot, \cdot)$ and relation
\eqref{4.S3}, we have
\begin{equation} \label{4.S4}
\begin{aligned}
 B(u,u)-B(v^\omega, v^\omega)
&= B(u-v^\omega,u)+B(v^\omega,u)-B(v^\omega, v^\omega)  \\
&= B(z,u)+B(v^\omega,u-v^\omega)\\
&=B(z,u)+B(u-z,z)\\
&= B(z,u)+B(u,z)-B(z,z).
\end{aligned}
\end{equation}
Similarly,
\begin{gather}
 B(b,b)-B(\xi^\omega, \xi^\omega)
= B(b,\zeta)+B(\zeta,b)-B(\zeta,\zeta), \label{4.S5} \\
 B(u,b)-B(v^\omega, \xi^\omega)
= B(z,b)+B(u,\zeta)-B(z,\zeta), \label{4.S6}\\
 B(b,u)-B(\xi^\omega, v^\omega)
= B(\zeta,u)+B(b,z)-B(\zeta,z). \label{4.S7}
\end{gather}
From \eqref{2.14}, \eqref{4.20} and \eqref{4.S4}-\eqref{4.S5}, we
find that $\check{z}(t)$ satisfies
\begin{gather}
\begin{aligned}
& \check{z}_t +\nu A\check{z}+\alpha^2A\check{z}_t
 +Q_\lambda(B(u,z)+B(z, u)-B(z,z)) \\
& -Q_\lambda(B(b,\zeta)+B(\zeta, b)-B(\zeta,\zeta))
 =0, \quad t>t_*,
\end{aligned} \label{4.67}\\
\check{z}(t_*)=Q_\lambda u(t_*). \label{4.68}
\end{gather}
Also, by \eqref{2.15}, \eqref{4.22} and \eqref{4.S6}-\eqref{4.S7},
we see that $\check{\zeta}(t)$ satisfies
\begin{gather}
\begin{aligned}
&\check{\zeta}_t +\mu A\check{\zeta}+\beta^2A\check{\zeta}_t
 +Q_\lambda(B(u,\zeta)+B(z, b)-B(z,\zeta)) \\
& -Q_\lambda(B(b,z)+B(\zeta, u)-B(\zeta,z))
 =0, \quad t>t_*,
\end{aligned}\label{4.69}\\
 \check{\zeta}(t_*)=Q_\lambda b(t_*). \label{4.70}
\end{gather}
Taking the inner product of equation \eqref{4.67} with $\check{z}$
 in $H$, we obtain
\begin{equation} \label{4.71}
\begin{aligned}
& \frac 12\frac{\mathrm{d}}{{\mathrm d}t}
 (\|\check{z}\|^2+\alpha^2\| \check{z}\|^2_1)
 +\nu\|\check{z}\|^2_1  \\
&\leqslant  |(B(\check{z}, u), \check{z})|
 +|(Q_\lambda(B(u, \overline{z})+B(\overline{z}, u)-B(z, \overline{z})), \check{z})|
  \\
&\quad  +|(Q_\lambda(B(b,\overline{\zeta})
 +B(\overline{\zeta}, b)
 -B(\overline{\zeta},\overline{\zeta})
 -B(\check{\zeta}, \overline{\zeta})
 -B(\overline{\zeta},\check{\zeta})),\check{z})|  \\
& \quad  -(Q_\lambda B(b,\check{\zeta}), \check{z})
 +|(B(\check{\zeta}, b), \check{z})|
 + (Q_\lambda B(\check{\zeta},\check{\zeta}),\check{z}).
\end{aligned}
\end{equation}
At the same time, taking the inner product of equation \eqref{4.69}
with $\check{\zeta}$ in $H$ gives
\begin{equation} \label{4.72}
\begin{aligned}
& \frac 12\frac{\mathrm{d}}{{\mathrm d}t}
 (\|\check{\zeta}\|^2+\beta^2\|\check{\zeta}\|^2_1)
 +\mu\|\check{\zeta}\|^2_1  \\
&\leqslant
 |(Q_\lambda(B(u, \overline{\zeta})+B(\overline{z}, b)
 -B(\overline{z}, \overline{\zeta})), \check{z})|  \\
&\quad  +|(Q_\lambda(-B(b,\overline{z})
 +B(\overline{\zeta}, u)
 -B(\overline{\zeta}, z)),\check{\zeta})|+|(B(\check{z}, b), \check{\zeta})|  \\
& \quad  +|(B(\check{z}, \overline{\zeta}), \check{\zeta})|
 -(Q_\lambda B(b, \check{z}), \check{\zeta})
 +|(B(\check{\zeta}, u), \check{z})|
 + (Q_\lambda B(\check{\zeta},\check{z}),\check{\zeta}).
\end{aligned}
\end{equation}
By the Poincar\'e inequality, we have
\begin{gather}
 \nu |\check{z}|^2_1
\geqslant
 \frac{\nu}{2}(\lambda_1\|\check{z}\|^2+|\check{z}|^2_1)
\geqslant
 \frac{\nu}{2(\lambda_1^{-1}+\alpha^2)}(\|\check{z}\|^2+\alpha^2|\check{z}|^2_1),
 \label{4.73}\\
 \mu |\check{\zeta}|^2_1
\geqslant
 \frac{\mu}{2}(\lambda_1\|\check{\zeta}\|^2+|\check{\zeta}|^2_1)
\geqslant
 \frac{\mu}{2(\lambda_1^{-1}+\beta^2)}
 (\|\check{\zeta}\|^2+\beta^2|\check{\zeta}|^2_1). \label{4.74}
\end{gather}
It then follows from \eqref{4.71}-\eqref{4.74} that
\begin{equation} \label{4.75}
\begin{aligned}
& \frac 12\frac{\mathrm{d}}{{\mathrm d}t}
 (\|\check{z}\|^2+\alpha^2\|\check{z}\|^2_1
 +\|\check{\zeta}\|^2+\beta^2\| \check{\zeta}\|^2_1)\\
&+\frac{\delta_4\delta_6}{2}(\|\check{z}\|^2+\alpha^2\|\check{z}\|^2_1
 +\|\check{\zeta}\|^2+\beta^2\| \check{\zeta}\|^2_1) \\
&\leqslant  |(B(\check{z}, u), \check{z})|
 +|(B(\check{\zeta}, b), \check{z})|
 +|(B(\check{z}, b), \check{\zeta})|
 +|(B(\check{z}, \overline{\zeta}), \check{\zeta})|\\
&\quad +|(B(\check{\zeta}, u), \check{z})|   +\Psi (t),
\end{aligned}
\end{equation}
where
\begin{equation} \label{4.76}
\begin{aligned}
 \Psi(t)
&=
 |(Q_\lambda(B(u, \overline{z})+B(\overline{z}, u)-B(z,\overline{z})), \check{z})| \\
&\quad +|(Q_\lambda(B(b,\overline{\zeta})
 +B(\overline{\zeta}, b)
 -B(\overline{\zeta},\overline{\zeta})
 -B(\check{\zeta}, \overline{\zeta})
 -B(\overline{\zeta},\check{\zeta})),\check{z})| \\
&\quad +|(Q_\lambda(B(u, \overline{\zeta})+B(\overline{z}, b)
 -B(\overline{z}, \overline{\zeta})), \check{z})|  \\
&\quad +|(Q_\lambda(-B(b,\overline{z})
 +B(\overline{\zeta}, u)
 -B(\overline{\zeta}, z)),\check{\zeta})|.
\end{aligned}
\end{equation}
Now the first five terms on the right-hand side of \eqref{4.75} can
be estimated as follows. Using \eqref{2.1}, Lemma \ref{L2.3} and
\eqref{4.5}-\eqref{4.6}, we obtain for $t\geqslant t_*$,
\begin{gather}
 |(B(\check{z}(t), u(t)), \check{z}(t))|
\leqslant
 \frac{c(\Omega)}{\lambda_1^{3/4}}|u(t)|_1\|\check{z}(t)\|^{1/2}|\check{z}(t)|^{3/2}_1
\leqslant
 \frac{c(\Omega)M_1}{\alpha\lambda_1^{3/4}\lambda^{1/4}}|\check{z}(t)|^{2}_1,
\label{4.77}\\
\begin{aligned}
 |(B(\check{z}(t), b(t)), \check{\zeta}(t))|
&\leqslant  \frac{c(\Omega)}{\lambda_1^{3/4}}
 |b(t)|_1  \|\check{z}(t)\|^{1/2}
 |\check{z}(t)|^{1/2}_1  |\check{\zeta}(t)|_1  \\
&\leqslant  \frac{c(\Omega)M_1}{\beta\lambda_1^{3/4}\lambda^{1/4}}
 |\check{z}(t)|_1|\check{\zeta}(t)|_1 \\
&\leqslant  \frac{c(\Omega)M_1}{\beta\lambda_1^{3/4}\lambda^{1/4}}
 (|\check{z}(t)|_1^2+|\check{\zeta}(t)|^2_1),
\end{aligned} \label{4.78}\\
 |(B(\check{\zeta}(t), b(t)), \check{z}(t))|
\leqslant  \frac{c(\Omega)M_1}{\beta\lambda_1^{3/4}\lambda^{1/4}}
 (|\check{z}(t)|_1^2+|\check{\zeta}(t)|^2_1),
\label{4.79}\\
\begin{aligned}
 |(B(\check{z}, \overline{\zeta}), \check{\zeta})|
&\leqslant  \frac{c(\Omega)}{\lambda_1^{3/4}}
 | \overline{\zeta} (t)|_1  \|\check{z}(t)\|^{1/2}
 |\check{z}(t)|^{1/2}_1  |\check{\zeta}(t)|_1  \\
&\leqslant  \frac{c(\Omega)(M_1/\beta+M^{(2)}_2)}{\lambda_1^{3/4}\lambda^{1/4}}
 |\check{z}(t)|_1|\check{\zeta}(t)|_1  \\
&\leqslant  \frac{c(\Omega)(M_1/\beta+M^{(2)}_2)}{\lambda_1^{3/4}\lambda^{1/4}}
 (|\check{z}(t)|_1^2+|\check{\zeta}(t)|^2_1),
\end{aligned} \label{4.80}\\
\begin{aligned}
 |(B(\check{\zeta}(t), u(t)), \check{z}(t))|
&\leqslant  \frac{c}{\lambda_1^{3/4}}
 |u(t)|_1  \|\check{\zeta}(t)\|^{1/2}
 |\check{\zeta}(t)|^{1/2}_1
 |\check{z}(t)|_1  \\
&\leqslant  \frac{c(\Omega)M_1}{\alpha\lambda_1^{3/4}\lambda^{1/4}}
 |\check{z}(t)|_1|\check{\zeta}(t)|_1 \\
&\leqslant  \frac{c(\Omega)M_1}{\alpha\lambda_1^{3/4}\lambda^{1/4}}
 (|\check{z}(t)|_1^2+|\check{\zeta}(t)|^2_1).
\end{aligned}\label{4.81}
\end{gather}
Inequalities \eqref{4.77}-\eqref{4.81} yield
\begin{equation} \label{4.82}
\begin{aligned}
& |(B(\check{z}, u), \check{z})|
 +|(B(\check{\zeta}, b), \check{z})|
 +|(B(\check{z}, b), \check{\zeta})|
 +|(B(\check{z}, \overline{\zeta}), \check{\zeta})|
 +|(B(\check{\zeta}, u), \check{z})|  \\
&\leqslant
 \frac{c(\Omega,\alpha,\beta,M_1,M^{(2)}_2)}{\lambda_1^{3/4}\lambda^{1/4}}
 (\|\check{z}(t)\|^2+\alpha^2|\check{z}(t)|_1^2
 +\|\check{\zeta}(t)\|^2
 +\beta^2|\check{\zeta}(t)|^2_1).
\end{aligned}
\end{equation}
Inserting \eqref{4.82} into \eqref{4.75} gives
\begin{equation} \label{4.83}
\begin{aligned}
&\frac 12\frac{\mathrm{d}}{{\mathrm d}t}
 \big(\|\check{z}\|^2+\alpha^2\|\check{z}\|^2_1
 +\|\check{\zeta}\|^2+\beta^2\| \check{\zeta}\|^2_1\big)  \\
& +\Big(\frac{\delta_4\delta_6}{2}-\frac{c(\Omega,\alpha,\beta,M_1,M^{(2)}_2)}
 {\lambda_1^{3/4}\lambda^{1/4}}\Big)
 (\|\check{z}\|^2+\alpha^2|\check{z}|^2_1
 +\|\check{\zeta}\|^2+\beta^2|\check{\zeta}|^2_1)\\
&\leqslant \Psi(t).
\end{aligned}
\end{equation}
Note that we can choose $\lambda$ large enough such that
\begin{align}\label{4.84}
 \frac{\delta_4\delta_6}{2}-\frac{c(\Omega,\alpha,\beta,M_1,M^{(2)}_2)}
 {\lambda_1^{3/4}\lambda^{1/4}}
 >0.
\end{align}
At the same time, employing the relations \eqref{4.63}-\eqref{4.64}
and the fact that $u$ and $b$ are bounded in the $V$ norm, we can
conclude from \eqref{4.76} that
\begin{align}\label{4.85}
 \lim_{t\to+\infty}\Psi(t)=0.
\end{align}
It then follows from \eqref{4.83}-\eqref{4.85} and the Gronwall
inequality that
\[
 \lim_{t\to+\infty}(\|\check{z}\|^2+\alpha^2|\check{z}|^2_1
 +\|\check{\zeta}\|^2+\beta^2| \check{\zeta}|^2_1)
 =0,
\]
for $\lambda$ large enough, satisfying
\begin{align}\label{4.86}
 \lambda>(\frac{c(\Omega,\alpha,\beta,M_1,M^{(2)}_2)}
{\delta_4\delta_6})^4\lambda_1^{-3}.
\end{align}
Therefore, \eqref{4.65} and \eqref{4.66} are proved for $\lambda$
large enough, satisfying relations \eqref{4.48}, \eqref{4.56} and
\eqref{4.86}. The proof of Theorem \ref{T4.1} is complete.
\end{proof}

\subsection*{Acknowledgments}
This research was supported by the NSFC of China  (No. 11271290) and
GSPT of Zhejiang Province (No.2014R424062).

The authors warmly thank the anonymous
referee for his/her careful reading of the manuscript and some
pertinent remarks that lead to various improvements to this paper.

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\end{document}